Definition 15.124.5. Let $R$ be a domain.

1. We say $R$ is a Bézout domain if every finitely generated ideal of $R$ is principal.

2. We say $R$ is an elementary divisor domain if for all $n , m \geq 1$ and every $n \times m$ matrix $A$, there exist invertible matrices $U, V$ of size $n \times n, m \times m$ such that

$U A V = \left( \begin{matrix} f_1 & 0 & 0 & \ldots \\ 0 & f_2 & 0 & \ldots \\ 0 & 0 & f_3 & \ldots \\ \ldots & \ldots & \ldots & \ldots \end{matrix} \right)$

with $f_1, \ldots , f_{\min (n, m)} \in R$ and $f_1 | f_2 | \ldots$.

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